Note the prediction for the power spectrum is larger by a factor of \(\epsilon ^{-1}\)  than the predictions for scalar fluctuations obtained by quantizing Einstein's Lagrangian[1]}. The reason for this is that quantum fluctuations of the scale factor in comoving gauge, treated as a quantum field, are suppressed by a factor of \(\sqrt{\epsilon }\)  relative to the
predictions of generic statistical mechanics, while black hole fluctuations \(\frac{\delta m}{m}\)  are independent of \(\epsilon \) . Note also that the factor of \(H\)  in the fluctuations is \(H_I\)  rather than the slow roll Hubble factor \(H(t)\) . Again this is because black holes are isolated quantum systems, whose fluctuations do not depend on the scale factor of the geometry in which they're embedded.
